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SOLVING SYSTEMS OF QUADRATIC EQUATIONS VIA EXPONENTIAL-TYPE GRADIENT DESCENT ALGORITHM

Meng Huang1,2, Zhiqiang Xu1,2   

  1. 1. LSEC, Inst. Comp. Math., Academy of Mathematics and System Science, Chinese Academy of Sciences, Beijing 100190, China;
    2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
  • Received:2018-06-01 Revised:2019-01-21 Online:2020-07-15 Published:2020-07-15

Meng Huang, Zhiqiang Xu. SOLVING SYSTEMS OF QUADRATIC EQUATIONS VIA EXPONENTIAL-TYPE GRADIENT DESCENT ALGORITHM[J]. Journal of Computational Mathematics, 2020, 38(4): 638-660.

We consider the rank minimization problem from quadratic measurements, i.e., recovering a rank r matrix X ∈ Rn×r from m scalar measurements y#em/em#=a#em/em#?XX?a#em/em#, a#em/em# ∈ Rn, #em/em#=1, …, m. Such problem arises in a variety of applications such as quadratic regression and quantum state tomography. We present a novel algorithm, which is termed exponential-type gradient descent algorithm, to minimize a non-convex objective function f(U)=1/4m Σ#em/em#=1m (y#em/em#-a#em/em#?UU?a#em/em#)2. This algorithm starts with a careful initialization, and then refines this initial guess by iteratively applying exponential-type gradient descent. Particularly, we can obtain a good initial guess of X as long as the number of Gaussian random measurements is O(nr), and our iteration algorithm can converge linearly to the true X (up to an orthogonal matrix) with m=O (nr log(cr)) Gaussian random measurements.

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